About 20 July, York was in Bedford and Bolingbroke in Leicester. You may need a map at this point, but those two places are a fair few miles apart. Henry had visited most of his northern estates and recruited large numbers of followers, mainly his tenants and retainers. At Doncaster he had been joined by the earls of Northumberland and Westmorland, an unusual case of the Nevilles and the Percies being on the same side. It appears that there he swore a solemn oath that he was not interested in taking Richard's crown. This must have been comforting to Westmorland and Northumberland as there was plenty of time for things to go pear-shaped, and at least they would be able to swear blind that they had no intention of deposing the King. Northumberland was eventually to swear exactly that, as justification for taking up arms against Henry IV on three separate occasions!
In addition, according to the Chronicler Creton, Archbishop Arundel was preaching a crusade against Richard II and claiming he had a Papal Bull to that effect! He was lying through his teeth - indeed technically he was not even an archbishop, having been deprived by the Pope. Not that you will pick this last detail up from your average Richard II textbook...
The likelihood is that the Duke of York picked up news of Henry's strength, and decided he couldn't meet him in battle. Anyway, he moved to Oxford, where it appears certain people deserted him. These were the earl of Wiltshire, and Bushey and Green of the King's council. It's possible they left with his agreement as he seems to have decided to move Queen Isabelle from Windsor to Wallingford, the latter being considered safer, and someone would have had to organise this. To cut a long story short, these three ended up in Bristol Castle. Bristol being a port and all that they may have hoped, or even have expected, that Richard would land there.
York meanwhile moved on into Gloucestershire. Henry had picked up on his change of direction and tracked him, moving from Leicester to Coventry, Warwick and Evesham. (I told you you might need that map.) On 24 July Richard landed in Wales.
York sent some of his people into Wales to meet the King, but for whatever reason he didn't follow himself with what was left of his army. He had suffered desertions, morale was probably low, and Bolingbroke was now uncomfortably close. None of these reasons seem to be a particularly good argument for a further change of direction, this time towards Bristol. If there was a plan at this point it can only have been to hole up in Bristol with those who had already made their way there ahead of him.
Instead, Bolingbroke caught up with his uncle outside Berkeley Castle on 27 July. There was actually some fighting because Bishop Despenser and Sir William Elmham were captured and disarmed. However, in view of Henry's superior numbers there was no realistic alternative to capitulation, this taking place in Berkeley parish church. The Chronicler Walsingham says that York was reluctant to stand in the way of a nephew who had come for his rightful inheritance. This may be a rare case of Walsingham telling the truth. In any event, York had painted himself into a corner, and there was not really anything else he could have done at this point. (Suicidal charges were not in the family tradition at this time.)